jmc

There are 12 hours on a clock, so each hour mark is $360^{\circ }/12=30^{\circ }$ from its neighbors. At 2:48 the minute hand points at minute 48, which is $\frac{3}{5}$ of the way from hour 9 to hour 10. Therefore, the minute hand is $\frac{3}{5}\cdot 30=18^{\circ }$ past hour 9, which means it is $30^{\circ }-18^{\circ }=12^{\circ }$ shy of hour 10. This means the minute hand is $2\cdot 30^{\circ }+12^{\circ }=72^{\circ }$ shy of hour 12.

The hour hand is $\frac{48}{60}=\frac{4}{5}$ of the way from hour 2 to hour 3, so it is $\frac{4}{5}\cdot 30^{\circ }=24^{\circ }$ past hour 2. So, the hour hand is $2\cdot 30^{\circ }+24^{\circ }=84^{\circ }$ past hour 12.

Combining the angles between each hand and hour 12, the angle between the hands is $72^{\circ }+84^{\circ }=\boxed{156^{\circ }}$.